Surface temperature distribution along a thin liquid layer due to thermocapillary convection

The surface temperature distributions due to thermocapillary convections in a thin liquid layer with heat fluxes imposed on the free surface are investigated. The nondimensional analysis predicts that, when convection is important, the characteristic length scale in the flow direction L, and the characteristic temperature difference ΔT₀, can be represented by and , respectively, where L_R and ΔT_T are the reference scales used in the conduction-dominant situations with A denoting the aspect ratio and Ma the Marangoni number. Having had L and ΔT₀ defined, the global surface-temperature gradient (), the global thermocapillary driving-force, and other interesting features can then be readily determined. Finally, numerical calculations involving a Gaussian heat flux distribution are presented to justify these two relations.